bvartools (version 0.0.1)

fevd: Forecast Error Variance Decomposition

Description

Produces the forecast error variance decomposition of a Bayesian VAR model.

A plot function for objects of class "bvarfevd".

Usage

fevd(object, response = NULL, n.ahead = 5, type = "oir",
  normalise_gir = FALSE)
# S3 method for bvarfevd
plot(x, ...)

Arguments

object

an object of class "bvar", usually, a result of a call to bvar or bvec_to_bvar.

response

name of the response variable.

n.ahead

number of steps ahead.

type

type of the impulse responses used to calculate forecast error variable decompositions. Possible choices are orthogonalised oir (default) and generalised gir impulse responses.

normalise_gir

logical. Should the GIR-based FEVD be normalised?

x

an object of class "bvarfevd", usually, a result of a call to fevd.

...

further graphical parameters.

Value

A time-series object of class "bvarfevd".

Details

The function produces forecast error variance decompositions (FEVD) for the VAR model $$y_t = \sum_{i = 1}^{p} A_{i} y_{t-i} + A_0^{-1} u_t,$$ with \(u_t \sim N(0, \Sigma)\).

If the FEVD is based on the orthogonalised impulse resonse (OIR), the FEVD will be calculated as $$\omega^{OIR}_{jk, h} = \frac{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i P e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma \Phi_i^{\prime} e_j )},$$ where \(\Phi_i\) is the forecast error impulse response for the \(i\)th period, \(P\) is the lower triangular Choleski decomposition of the variance-covariance matrix \(\Sigma\), \(e_j\) is a selection vector for the response variable and \(e_k\) a selection vector for the impulse variable.

If type = "sir", the structural FEVD will be calculated as $$\omega^{SIR}_{jk, h} = \frac{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} \Sigma^{\frac{1}{2}} e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} \Sigma A_0^{-1\prime} \Phi_i^{\prime} e_j )},$$ where \(\sigma_{jj}\) is the diagonal element of the \(j\)th variable of the variance covariance matrix.

If type = "gir", the generalised FEVD will be calculated as $$\omega^{GIR}_{jk, h} = \frac{\sigma^{-1}_{jj} \sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma \Phi_i^{\prime} e_j )},$$ where \(\sigma_{jj}\) is the diagonal element of the \(j\)th variable of the variance covariance matrix.

If type = "sgir", the structural generalised FEVD will be calculated as $$\omega^{SGIR}_{jk, h} = \frac{\sigma^{-1}_{jj} \sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} \Sigma e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} \Sigma A_0^{-1\prime} \Phi_i^{\prime} e_j )}$$.

Since GIR-based FEVDs do not add up to unity, they can be normalised by setting normalise_gir = TRUE.

References

L<U+00FC>tkepohl, H. (2007). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Pesaran, H. H., & Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models. Economics Letters, 58, 17-29.

Examples

# NOT RUN {
data("e1")
e1 <- diff(log(e1))

data <- gen_var(e1, p = 2, deterministic = "const")

y <- data$Y[, 1:73]
x <- data$Z[, 1:73]

set.seed(1234567)

iter <- 500 # Number of iterations of the Gibbs sampler
# Chosen number of iterations should be much higher, e.g. 30000.

burnin <- 100 # Number of burn-in draws
store <- iter - burnin

t <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
m <- k * nrow(x) # Number of estimated coefficients

# Set (uninformative) priors
a_mu_prior <- matrix(0, m) # Vector of prior parameter means
a_v_i_prior <- diag(0, m) # Inverse of the prior covariance matrix

u_sigma_df_prior <- 0 # Prior degrees of freedom
u_sigma_scale_prior <- diag(0, k) # Prior covariance matrix
u_sigma_df_post <- t + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
u_sigma_i <- diag(.00001, k)
u_sigma <- solve(u_sigma_i)

# Data containers for posterior draws
draws_a <- matrix(NA, m, store)
draws_sigma <- matrix(NA, k^2, store)

# Start Gibbs sampler
for (draw in 1:iter) {
  # Draw conditional mean parameters
  a <- post_normal(y, x, u_sigma_i, a_mu_prior, a_v_i_prior)

# Draw variance-covariance matrix
u <- y - matrix(a, k) %*% x # Obtain residuals
u_sigma_scale_post <- solve(u_sigma_scale_prior + tcrossprod(u))
u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k)
u_sigma <- solve(u_sigma_i) # Invert Sigma_i to obtain Sigma

# Store draws
if (draw > burnin) {
  draws_a[, draw - burnin] <- a
  draws_sigma[, draw - burnin] <- u_sigma
  }
}

# Generate bvar object
bvar_est <- bvar(y = y, x = x, A = draws_a[1:18,],
                 C = draws_a[19:21, ], Sigma = draws_sigma)

# Generate forecast error variance decomposition (FEVD)
bvar_fevd <- fevd(bvar_est, response = "cons")

# Plot FEVD
plot(bvar_fevd, main = "FEVD of consumption")
data("e1")
e1 <- diff(log(e1))

data <- gen_var(e1, p = 2, deterministic = "const")

y <- data$Y[, 1:73]
x <- data$Z[, 1:73]

set.seed(1234567)

iter <- 500 # Number of iterations of the Gibbs sampler
# Chosen number of iterations should be much higher, e.g. 30000.

burnin <- 100 # Number of burn-in draws
store <- iter - burnin

t <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
m <- k * nrow(x) # Number of estimated coefficients

# Set (uninformative) priors
a_mu_prior <- matrix(0, m) # Vector of prior parameter means
a_v_i_prior <- diag(0, m) # Inverse of the prior covariance matrix

u_sigma_df_prior <- 0 # Prior degrees of freedom
u_sigma_scale_prior <- diag(0, k) # Prior covariance matrix
u_sigma_df_post <- t + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
u_sigma_i <- diag(.00001, k)
u_sigma <- solve(u_sigma_i)

# Data containers for posterior draws
draws_a <- matrix(NA, m, store)
draws_sigma <- matrix(NA, k^2, store)

# Start Gibbs sampler
for (draw in 1:iter) {
  # Draw conditional mean parameters
  a <- post_normal(y, x, u_sigma_i, a_mu_prior, a_v_i_prior)

# Draw variance-covariance matrix
u <- y - matrix(a, k) %*% x # Obtain residuals
u_sigma_scale_post <- solve(u_sigma_scale_prior + tcrossprod(u))
u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k)
u_sigma <- solve(u_sigma_i) # Invert Sigma_i to obtain Sigma

# Store draws
if (draw > burnin) {
  draws_a[, draw - burnin] <- a
  draws_sigma[, draw - burnin] <- u_sigma
  }
}

# Generate bvar object
bvar_est <- bvar(y = y, x = x, A = draws_a[1:18,],
                 C = draws_a[19:21, ], Sigma = draws_sigma)

# Generate forecast error variance decomposition (FEVD)
bvar_fevd <- fevd(bvar_est, response = "cons")

# Plot FEVD
plot(bvar_fevd, main = "FEVD of consumption")
# }